Finally, we evaluate `PH' on IBM's 16-qubit Melbourne chip with 8 QAOA MaxCut programs.
We generate 4 regular graphs of 7 to 10 nodes with 4 edges per node (`REG-n(7-10)-d4'), and 4 random graphs of 7 to 10 nodes with edge probability 0.5 (`RD-n(7-10)-p0.5').
We prepare 1-level QAOA circuits on these graphs and then optimize the parameters in the simulator.
Those circuits with the optimized parameters are then evaluated on the Melbourne chip (40960 shots per circuit).
The baseline is `Qiskit_L3' with the Pauli strings ordered by iterating over the adjacency matrix (Qiskit default configuration).
Figure REF  shows the improvement of the success probability after applying `PH' optimizations. The `Estimated Success Probability' (ESP), a widely used metric in guiding the compiler optimization [1]}, [2]}, [3]}, is a theoretical estimation of the success probability based on the program and the hardware noise model.
The `Real System Success Probability' (RSP) is the number of trials with correct measurement results divided by the total number of trials when executing on the real machine.
Applying `PH' can improve the ESP by \(2.11\times \)  on average (up to \(3.00\times \) ) based on the noise model of the tested device, by reducing the CNOT count and circuit depth by 15.1% and 36.2%, respectively on average.
On the real machine, `PH' can improve the RSP by \(1.24\times \)  on average (up to \(1.87\times \) ).
There is a gap between the results from ESP and RSP because the noise model only provides limited hardware information.
We expect that the compilation can be further improved with more detailed hardware models.
<FIGURE>